Net intensity for interference from incoherent sources

Question

Why do we simply add the intensities for interference from two incoherent sources? What's the proof for that? I know that incoherent means that their phase difference changes with time

Answer 1

Imagine that you had two sources emitting waves of exactly the same amplitude $A$, same frequency with a constant phase difference.

Where there is superposition of these wave there will be an interference pattern.

At some positions the waves from the two sources will arrive in phase with one another and the resulting amplitude will be $A+A = 2A$ with an intensity proportional to $(2A)^2 = 4A^2$.
At other positions the waves will arrive exactly out of phase with one another and the resultant amplitude will be $A-A =0$.
So the intensity in the region of interference will range from zero to being proportional to $4A^2$.
The average intensity will be proportional to $2A^2$ with is the sum of the intensities due to the individual sources.
So interference does not destroy or create energy rather it channels the flow of different amounts of energy in different directions.

Now with the two same source let the phase of one source relative to the other change with time.

What will be observed?

It will be a moving interference pattern and at any one position the intensity will vary for zero to being proportional to $4A^2$ but on average at that position the intensity will be proportional to $2A^2$, the sum of the intensities of the two individual sources.

You can perhaps now extend this arguement to incoherent sources where there would be superposition and at any instant of time an interference pattern but in the next instant of time the interference pattern would have moved an so at any position the intensity would average out as the sum of the ontensities from the two sources.